arXiv research

Proves local solvability for $G_2$-structures with Poisson equations.

Solutions to a differential equation link to contact structures.

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more…

Paper discovers structural dynamics equations from only acceleration data.

Formula connects $G_2$-structure geometry to Poisson equation.

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex s…

Superintegrable systems on surfaces are classified geometrically.

New equations for Cosserat media motions derived from bundle automorphisms.

Global invariant for path structures and differential equations defined on torus.

In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equa…

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.

Study on G2 structures with torsion and existence of solutions.

Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.

Monograph explores algebraic structures related to Yang-Baxter equation.

We discuss relations between the para-CR structures and differential equations (both ODEs and PDEs of finite type).

In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g >= 1, gives rise to a family of moduli spaces carrying a hy…

We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structu…

Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.

In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as $i_{\dot γ}dσ=0$, where $σ$ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where $σ$ is a differen…

The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.

Study solves complex Hessian equation on Hermitian manifolds.

We apply the technique of integrable extensions to the symmetry pseudo-group of the dKP-hyper CR interpolating equation. This allows us to find a covering for this equation and to construct multi-valued Einstein-Weyl structures.

The paper studies hyperkähler structures and adapted complex structures using the Monge-Ampère equation.

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely…

New framework tackles geometric structure existence and classification.

Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…

We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which …

Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

The paper studies Einstein-type manifolds with structural conditions.

Affine manifolds linked to integrable equations and geometric structures.

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

Solves geometric problems using fully nonlinear equations and Morse theory.

Study proves structure results for homogeneous spaces supporting specific equations.

New conformal geometry method solves Einstein-Weyl equations.

We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in …

The hyper-CR Einstein-Weyl structures on $\R^3$ can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Krugliko…

We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.

Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…

Motivated by the geospin matrix $W$ as a new variable in Riemannian geometry. Then, we use four real dynamical variables $\left\{ a,α,v,W \right\}$ to show the dynamical essence of Cartan structural equation, we obtain the geometrodynamics on Riemannian manifolds that can be expressed below \begin{align} & Θ/d{{t}^{2}}…

We consider Riemannian 4-manifolds $(X,g_X)$ with a Spin^c-structure and a suitable circle bundle $Y$ over $X$ such that the Spin^c-structure on $X$ lifts to a spin structure on $Y$. With respect to these structures a spinor $φ$ on $X$ lifts to an untwisted spinor $ψ$ on $Y$ and a U(1)-gauge field $A$ for the Spin^c-st…

Paper solves Hessian equations on Kähler manifolds.

In \cite{LZ2} it is proved that for certain class of perturbations of the hyperbolic equation $u_t=f(u) u_x$, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations posse…

Graphical notation simplifies complex polynomial constraints in linear models.

The paper proves isometric embedding equations in low Sobolev regularity.